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For propagation perpendicular to the equilibrium magnetic field, this plasma wave has its electric field perpendicular to that magnetic field and generally both parallel and perpendicular to its wavevector. Its dispersion relation is $n^2=RL/S$, using the magnetized cold-plasma dielectric tensor conventions $R=S-g$, $L=S+g$.

Use Gaussian units, plane waves proportional to $e^{i\mathbf{k}\cdot \mathbf{r}-i\omega t}$, and the signed cyclotron frequency ${\Omega }_s=q_s{B}_0/(m_{s}c)$. Write ${\Omega }_e=e{B}_0/(m_{e}c)>0$ for its electron magnitude, so the signed electron value is $-{\Omega }_e$. Define the species plasma frequency by ${\omega }_{ps}^2=4\pi n_{0s}{q}_s^2/m_s$. The linearization of the cold Lorentz force equations givesSolving this linear system and using the species current density ${\mathbf{j}}_s=n_{0s}{q}_s{\mathbf{u}}_s$ gives the conductivity tensorThe magnetized cold-plasma dielectric tensor is $\mathbf{ϵ}=\mathbf{I}+(4\pi i/\omega ){\sum }_s{\mathbf{\sigma }}_s$. ThusAt the stated electron frequency scale, the ion contribution is smaller by $O(Z{m}_e/m_i)$ for $q_i=Ze$ and quasineutrality. To this accuracy,These expressions require negligible collisions and the cold ordering $k{v}_{\mathrm{th},s}\ll |\omega |$; sufficiently close to a cyclotron resonance, thermal broadening or collisions must also be retained.

A plasma wave resonance has $n^2\to \infty$. At finite coefficients of the magnetized cold-plasma dielectric tensor, its generic condition is the vanishing of the coefficient of $n^4$:A plasma wave cutoff has $n^2=0$, so the constant term of the dispersion relation vanishes:Both statements are branch conditions: check the undivided dispersion relation at special angles and simultaneous zero factors. In particular, the parallel plasma oscillation at $P=0$ is a longitudinal branch with arbitrary $k$ in the cold limit, while the transverse parallel branches have plasma wave cutoffs at $R=0$ or $L=0$. Poles of the magnetized cold-plasma dielectric tensor can give additional limiting plasma wave resonances, such as the parallel cold-plasma right-handed wave at $\omega ={\Omega }_e$; these are not obtained by assuming finite $S,P,g$.

An electrostatic plasma wave has $\delta \mathbf{E}=-i\mathbf{k}\varphi$. In the absence of externally imposed charge density, the Gauss law for the magnetized cold-plasma dielectric tensor is $\mathbf{k}\cdot \mathbf{ϵ}\cdot \delta \mathbf{E}=0$. Consequently,The antisymmetric terms of the magnetized cold-plasma dielectric tensor cancel in this scalar contraction. For oblique propagation this is the leading longitudinal condition in the large-refractive index approximation; a small transverse electric field is generally required by the full Maxwell equations at finite $n$. For parallel propagation an exactly longitudinal solution exists: $P=0$, hence ${\omega }^2\simeq {\omega }_{pe}^2$. It coincides with the $\theta \to 0$ limit of the finite-coefficient plasma wave resonance condition in c. At exactly $\theta =0$, $P$ factors out of the transverse dispersion relation, leaving the independent plasma oscillation rather than an isolated divergence of the transverse branches.

Write the ordinary plasma wave asThe eikonal function $S$ is dimensionless; it varies by order ${\epsilon }^{-1}$ across a background scale $L_{\mathrm{bg}}$, where $\epsilon \sim \lambda /L_{\mathrm{bg}}\ll 1$. The amplitudes and equilibrium plasma parameters vary on $L_{\mathrm{bg}}$, so $|\nabla {\mathbf{a}}_j|\ll k|{\mathbf{a}}_j|$ and $|\nabla k|\ll k^2$. The equilibrium is stationary, so $\omega$ is constant along propagation. Assume infinitesimal fields, negligible collisions, $k{v}_{\mathrm{th},s}\ll \omega$, negligible ion response, and a fixed ordinary plasma wave polarization with ${\mathbf{B}}_0\parallel \hat{\mathbf{z}}$ and spatial variation along $x$. The WKB approximation is used away from wave turning points and plasma wave resonances.

Replacing derivatives on the fast factor by $i\mathbf{k}$ and $-i\omega$ in the Maxwell equations reproduces the local magnetized cold-plasma dielectric tensor equation of b. In this one-dimensional ordinary plasma wave geometry, the Maxwell equations and the local cold parallel current density in fact give the scalar equation directly:For $\delta E_z=a(x)e^{iS(x)}$, substitution givesThe leading eikonal equation is $(S^{\prime }{)}^2=k^2(x)$, and the next order gives $2k{a}^{\prime }+k^{\prime }a=0$, hence the flux-normalized WKB amplitude $a\propto k^{-1/2}$. In particular, require $|k^{\prime }|/|k{|}^2\ll 1$ and $|k^{\prime \prime }|/|k{|}^3\ll 1$; these fail at the wave turning points.

A point on a plasma wave branch where the local wavenumber tends to zero at finite frequency. The mode may change between propagation and evanescence. For a finite magnetized cold-plasma dielectric tensor, candidate cutoffs satisfy $PRL=0$; special propagation angles require checking which factor belongs to the chosen branch.

A limiting point of a plasma wave branch with divergent refractive index. At finite coefficients of the magnetized cold-plasma dielectric tensor, the generic condition is $S{\sin }^2\theta +P{\cos }^2\theta =0$. Poles of that magnetized cold-plasma dielectric tensor require a separate limiting analysis. Collisions, thermal dispersion or mode conversion can regularize a singular cold prediction.